The Sphere in Three Dimensions and Higher:
Generalizations and Special Cases

Matt Chisholm
matt@theory.org

Abstract:

I examine the general mathematical object that is the sphere. First, I examine several constructions general to all spheres. Then, I look at the Clifford Tori, the Hopf Fibration, group structure and surgery in $\Bbb{S}^{3}$, with respect to the unique structure of $\Bbb{S}^{3}$. Finally, I examine a unique dissection of $\Bbb{S}^{3}$ which takes advantage of a certain triangulation and the Clifford Tori. This dissection can be used to explain surgery on the unknot in an interesting way.